Primes in Divisibility Sequences
نویسندگان
چکیده
We give an overview of two important families of divisibility sequences: the Lehmer–Pierce family (which generalise the Mersenne sequence) and the elliptic divisibility sequences. Recent computational work is described, as well as some of the mathematics behind these sequences.
منابع مشابه
Primes in Elliptic Divisibility Sequences
Morgan Ward pursued the study of elliptic divisibility sequences initiated by Lucas, and Chudnovsky and Chudnovsky suggested looking at elliptic divisibility sequences for prime appearance. The problem of prime appearance in these sequences is examined here from a theoretical and a practical viewpoint. We exhibit calculations, together with a heuristic argument, to suggest that these sequences ...
متن کاملAlgebraic Divisibility Sequences over Function Fields
In this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field...
متن کاملPrime powers in elliptic divisibility sequences
Certain elliptic divisibility sequences are shown to contain only finitely many prime power terms. In some cases the methods prove that only finitely many terms are divisible by a bounded number of distinct primes.
متن کاملPrimes in Sequences Associated to Polynomials (after Lehmer)
In a paper of 1933, D.H. Lehmer continued Pierce’s study of integral sequences associated to polynomials, generalizing the Mersenne sequence. He developed divisibility criteria, and suggested that prime apparition in these sequences – or in closely related sequences – would be denser if the polynomials were close to cyclotomic, using a natural measure of closeness. We review briefly some of the...
متن کاملOn the Divisibility of Fibonacci Sequences by Primes of Index Two
Brother Alfred has characterized primes dividing every Fibonacci sequence [2] based on their period and congruence class mod 20. More recently, in [4] Ballot and Elia have described the set of primes dividing the Lucas sequence, meaning they divide some term of the sequence. Our purpose here is to extend the results of the former paper utilizing the methods of the latter. In particular we will ...
متن کامل